Conventional sampled control loop for controlling a process can be designed to provide an optimal response provided the consecutive sampling intervals for the loop as well as the time constants and propagation delays associated with the process are each constant. However, not all transducers for monitoring the output of a process produce outputs at constant time invervals. Furthermore, the time constants associated with a process may change dynamically as a function of load conditions or as a result of interaction with cooperating processes.
FIG. 1 illustrates a sampled control loop which is representative of the prior art. A sampling switch 10 opens and closes at a fixed sampling rate. A gain element 14 provides position feedback, a delay element 12 and a gain element 15 provide velocity feedback, and a delay element 13 and a gain element 16 provide acceleration feedback. This progression can be continued to provide jerk feedback, etc., for as many gain elements and delay elements as desired. A zero order hold circuit 18 converts the impulse output of a summing junction 17 to step-like signals which are more compatible with realistic systems such as motors. A linearizer 19 makes the process 7 look like a linear process if it is proportional but nonlinear. The gains of gain elements 14, 15, and 16 are dependent on the sampling rate, the scale factor and the time constants of the process 7. Gain elements 14, 15, and 16 can be determined experimentally or by a computer aided optimization technique. The control loop of FIG. 1 has the disadvantage that if the sampling rate changes, or if the scale factor of process 7 changes, or any of the time constants change, the response loop is no longer optimized. To maintain optimum response necessitates reoptimizing by the methods previously mentioned. Situations that do not allow the time necessary to reoptimize cannot be maintained at peak efficiency. Furthermore, such control loops cannot remain optimized in the face of dynamic changes in any of the aforementioned variables.
It would be desirable if a sampled control loop were designed to maintain a desired optimal response regardless of dynamic changes in the sampling interval for the loop, the time constant for the process, or the propagation delay for the transducer.
It can be shown for any particular type of loop response deemed optimal that there exist gain control functions corresponding to the gain elements 14, 15, 16, . . . These gain control functions have the loop sampling interval, process time constants and (possibly) transducer propagation delay, as independent variables. If each of gain elements 14, 15, 16, . . . , provided a gain in accordance with the value of its associated gain control function then the response of the loop would remain optimal even though there were dynamic changes in the independent variables for the functions.
In a preferred embodiment of the invention to be discussed below a gain optimizer responds dynamically to the aforementioned variables through approximating functions whose values control the degree of gain provided by gain elements corresponding to 14, 15, 16, . . .
The gain optimizer implements approximating functions rather than the actual gain control functions because the latter are extremely difficult, or in many cases impossible, to derive analytically. The analytical derivation of the actual gain control functions involves solving for the roots of a polynomial of degree one higher than the total number of independent variables allowed to vary dynamically. As is well known, no general closed form solutions exist for polynomials of degree five or higher. And even where the closed form solution for the polynomial does exist the manner in which the roots are used while finding the actual set of gain control functions is extremely cumbersome.
It is shown in connection with a preferred embodiment of the invention that a collection of approximating functions can be defined by operationalizing the definition of optimal response and then allowing the independent variables to assume a wide range of values. In this way is found a collection of numerical values that each actual gain control function ought to provide. These values can either be condensed into a look-up table for use in the gain optimizer, or modeled to a desired degree of accuracy by a compact set of approximating functions specifically choosen for that purpose.